In applications of statistics especially in biostatistics, independent two-sample data from case-control study for instance are common. If the two nonparametric underlying
distributions are linked in a certain parametric way, then we can find better estimates of the distributions by efficiently combining the two independent samples. Examples of such linked models are two-sample proportional odds model [1]}, two-sample proportional hazard model [2]}, two-sample density ratio (DR) model
[3]}, [4]}, [5]}, and so on.
Suppose that the densities \(f_0\)  and \(f_1\)  of “control” data \(X_0\)  and “case” data \(X_1\) , respectively, satisfy the following density ratio model
\(f_1(x)=f(x;\mathbf {\alpha })=f_0(x)\exp \lbrace \mathbf {\alpha }^\mathrm {\scriptscriptstyle \top }\tilde{r}(x)\rbrace ,\) 
